Self-dual Maps I : antipodality

TL;DR

The paper addresses when a self-dual map $G$ on the sphere is antipodally self-dual, i.e., when its dual $G^*$ is antipodally embedded with respect to $G$, formalized via $-\widehat{G}=\widehat{G}^*$. It develops a combinatorial criterion based on involutive labelings of the square graph $I(G)^{\square}$ to characterize antipodal self-duality (and links to strong involutivity), and proves that such labelings exist iff $I(G)^{\square}$ admits an involutive labeling without fixed vertices. It then constructs and analyzes several infinite families—$W_n$, $E_n$, and $P_n^{\ell}$—and an adhesion construction to generate antipodally self-dual maps, while also connecting these notions to antipodal symmetry of related graphs like $med(G)$ and $I(G)$. The results provide effective tools for testing antipodal self-duality and offer insights into symmetry and amphicheirality questions in related geometric and topological contexts.

Abstract

A self-dual map $G$ is said to be \emph{antipodally self-dual} if the dual map $G^*$ is antipodal embedded in $\mathbb{S}^2$ with respect to $G$. In this paper, we investigate necessary and/or sufficient conditions for a map to be antipodally self-dual. In particular, we present a combinatorial characterization for map $G$ to be antipodally self-dual in terms of certain \emph{involutive labelings}. The latter lead us to obtain necessary conditions for a map to be \emph{strongly involutive} (a notion relevant for its connection with convex geometric problems). We also investigate the relation of antipodally self-dual maps and the notion of \emph{ antipodally symmetric} maps. It turns out that the latter is a very helpful tool to study questions concerning the \emph{symmetry} as well as the \emph{amphicheirality} of \emph{links}.

Figures (20)

• Figure 1: A map and its dual, the squares graph, the vertex-face incidence graph and the medial.
• Figure 2: (Right) $I(G)$ (straight edges and black and vertices in white circles) and $med(G)$ (dashed edges and vertices in transparent squares) (Left) Graph $I(G)^\Box$.
• Figure 3: (Left) a self-dual map $G'$ (the isomorphism $\sigma$ is given by $\sigma(a)=A, \sigma(b)=B, \sigma(c)=C,$ etc.) not admitting a strongly involutive dual-isomorphism (Right) $I(G')$ admitting a symmetric cycle of length 8 (bold edges).
• Figure 4: (Left) A self-dual map $G$ (straight edges and black vertices) and $G^*$ (dashed edges and white vertices). It can easily be checked that $G$ do not admit a strongly involutive isomorphism. (Right) An involutive labeling of $I(G)^\square$ without fixed vertices.
• Figure 5: (Left) The self-dual map $W_4$ with its dual. (Right) $I^\square(W_4)$ together with an involutive labelling with two fixed vertices de type $V_E$: $\gamma=\bar{\gamma}$ and $\epsilon=\bar{\epsilon}$ (bold squares).

Theorems & Definitions (26)

• Theorem 1
• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Lemma 1
• proof
• Theorem 2
• proof
• Corollary 1